Rich people do crazy things with their money, but no one’s ever accused billionaire New York City mayor Michael Bloomberg of being the kind to throw money away on a lost cause. So some observers are confused by his $12 million ad buy during Congress’s Easter recess. Bloomberg’s ads push for the expansion of legally required background checks to include private sales, a move that close to 90% of Americans support, including 74% of NRA members, according to Republican pollster Frank Luntz.

Bloomberg’s not the only one pushing the issue this week. Backed by the mothers of victims of gun violence, an impassioned President Obama held an event at the White House on Thursday morning touting his multifaceted gun-control package and singling out the background-check issue in particular. “Ninety percent of Americans — 90% — support background checks,” Obama said. “How often do 90% of Americans agree on anything?” Groups allied with Bloomberg, like Gabby Giffords’ Americans for Responsible Solutions, are also launching a background-check blitz.

So why make a big push to influence public opinion when public opinion’s already on your side?

The answer lies in the stalled negotiations between Democratic Senator Chuck Schumer and Republican Senator Tom Coburn over gun legislation in Congress. How that standoff is resolved will determine whether the latest gun-control effort, launched in the wake of December’s killing of 20 children and 6 adults in Newtown, Conn., goes anywhere.

Federal law already requires commercial gun sellers to run a background check on a gun buyer before the sale of a weapon, and to keep a paper record of the sale. But sellers at gun shows and in private sales don’t have to do either. While increased background checks are popular, Coburn and the NRA say there is less support for expanding paper records, which gun-rights advocates warn could be used to create a national database of gun ownership that might allow the government to confiscate firearms. Gun-control groups, and Schumer, say any bill without the paper records would be meaningless.

The paper-record issue constitutes the last unresolved detail preventing broad Republican support for expanded background checks. Schumer had already agreed to a raft of other compromises, including:

Exempting holders of concealed-carry permits from background checks

Exempting intrafamily sales from the checks

Establishing an online portal for checks so that rural sellers don’t need to travel to comply with the law

Allowing vets diagnosed with mental illness to seek a reassessment of their case

Flexibility in who would retain the paper records (the private seller, a local licensed seller or even the gun manufacturer)

“We got nine-tenths of the way with Coburn,” says one senior Democratic Senate staffer. But the Oklahoma Republican argued that conservatives simply wouldn’t support a bill that expanded record keeping, even one that merely extended the existing system for commercial sales to private sales. Schumer said gun-control groups felt strongly that expanded background checks without record keeping would be meaningless. The two sides amicably agreed to test the support for those positions on both sides.

Which gets us back to the ad blitz. The break in negotiations “gives an opportunity for some of the groups that have been trying to work this, including [Bloomberg’s] group and Gabby Giffords’ group, to test their abilities to put pressure on lawmakers and to organize,” says the Democratic staffer. “This will be a little bit of a test of these groups’ muscle,” says the staffer.

It also explains why Obama spent so much time Thursday passionately urging everyday Americans to make their voices heard. “Speak up, we need your voices in this debate,” Obama said. If your representative is not one of the 90% supporting background checks, Obama said, “then you should ask them, ‘Why not? Why are you part of the 10%?'”

There are three possible outcomes to the test of wills. Bloomberg and his allies could successfully pressure Republicans to back down on paper records, likely giving Obama a win on two of the four measures he sought in the wake of Newtown (a bill increasing the penalties for gun trafficking is likely to pass, while an assault-weapons ban and an expensive bill boosting school-safety programs are expected to fail). Alternatively, if the Bloomberg effort fails to pressure conservatives to accept paper records for private sales, gun-control advocates will have to decide whether to accept expanded background checks without paper records or scrap them entirely.

The last outcome would be a failure all around, given the broad support for universal background checks, and the evidence that they could do something to diminish gun deaths. Bloomberg’s group has compiled records claiming that last year 6.6 million guns were sold privately without background checks. And a Bloomberg News article says that “a 2004 Bureau of Justice Statistics survey of inmates convicted of gun crimes found that 80 percent acquired the weapons through a private transfer.”